Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn real algebras generated by positive and nonnegative matrices
87
0
0.0
(
0
)
Authors
N.A. Kolegov
Ask ChatGPT about the research
No Arabic abstract
Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandi{c}, K. v{S}ivic (2017).
rate research
Read More
We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.
In this paper we give the relationship between the connected components of pure generalized Dynkin graphs and Nichols braided Lie algebras.
We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely the graph $P_1$ with one vertex and no edges. The graph $P_1$ has a unique greatest lower bound, namely the graph $P_2$ with two vertices and one directed edge. Our main result is a complete description of the greatest lower bounds of $P_2$; we call these graphs submaximal. We show that every graph that is not equivalent to $P_1$ and $P_2$ is below one of the submaximal graphs.
The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame $n$-slice algebras to the McKay quiver of a finite subgroup of $mathrm{GL}(n+1, mathbb C)$. In the case of $n=2$, we describe the relations for the $2$-slice algebras related to the McKay quiver of finite Abelian subgroups of $mathrm{SL}(3, mathbb C)$ and of the finite subgroups obtained from embedding $mathrm{SL}(2, mathbb C)$ into $mathrm{SL}(3,mathbb C)$.
Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce the irreducible tubal matrix, and provide two equivalent characterizations. Then, the celebrated Perron-Frobenius theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices can be generalized to the tubal matrix setting, while some are not. One reason is the defined positivity here has a different meaning to its usual sense. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
